﻿ 气体动理学统一算法的隐式方法研究
«上一篇
 文章快速检索 高级检索

 力学学报  2015, Vol. 47 Issue (5): 822-829  DOI: 10.6052/0459-1879-14-408 0

### 引用本文 [复制中英文]

[复制中文]
Mao Meiliang, Jiang Dingwu, Li Jin, Deng Xiaogang. STUDY ON IMPLICIT IMPLEMENTATION OF THE UNIFIED GAS KINETIC SCHEME[J]. Chinese Journal of Ship Research, 2015, 47(5): 822-829. DOI: 10.6052/0459-1879-14-408.
[复制英文]

### 文章历史

2014-12-19 收稿
2015-06-06 录用
2015–06–24 网络版发表

1. 中国空气动力研究与发展中心空气动力学国家重点实验室, 绵阳621000;
2. 中国空气动力研究与发展中心计算空气动力研究所, 绵阳621000;
3. 国防科学技术大学, 长沙410073

1 UGKS方法及其隐式实现

 ${f_t} + u{f_x} + v{f_y} = \frac{{{f^ + } - f}}{\tau }$ (1)

 $f^ + = f_{\rm M} + f^{ + + }$

 ${f_{\rm{M}}} = \rho {\left( {\frac{\lambda }{\pi }} \right)^{(K + 2)/2}}\exp \{ - \lambda \left[{{{\left( {u - U} \right)}^2} + {{\left( {v - V} \right)}^2} + {\xi ^2}} \right]\}$

 $\begin{array}{l} {\mathop \iiint \limits}{u_{\rm{n}}}{f_{\rm{p}}}\left( {\begin{array}{*{20}{c}} 1\\ u\\ v\\ {\frac{1}{2}({u^2} + {v^2} + {\xi ^2})} \end{array}} \right)dudvd\xi = \\ {\mathop \iint \limits}g\left( {\begin{array}{*{20}{c}} 1\\ u\\ v\\ {\frac{1}{2}({u^2} + {v^2})} \end{array}} \right){u_{\rm{n}}}du{\rm{d}}v + \\ \frac{1}{2}{\mathop \iint \limits}\left( {\begin{array}{*{20}{c}} 0\\ 0\\ 0\\ h \end{array}} \right){u_{\rm{n}}}du{\rm{d}}v \end{array}$

 $\frac{{\partial g_{i,j,l,m}^n}}{{\partial t}} + {u_l}\frac{{\partial g_{i,j,l,m}^n}}{{\partial x}} + {v_m}\frac{{\partial g_{i,j,l,m}^n}}{{\partial y}} = \frac{{(g_{i,j,l,m}^ + - {g_{i,j,l,m}})}}{{{\tau ^n}}}$ (2)

 $\begin{array}{l} (1 + \Delta t/{\tau ^n} + \Delta t \cdot {u_{l,m}}\nabla ){(\Delta g)_{l,m}} = \Delta t \cdot L_{i,j,l,m}^n\\ \;\;\;\;\;\;\;L_{i,j,l,m}^n = - {u_{l,m}}\nabla g_{l,m}^n + J_{l,m}^n = \\ \;\;\;\;\;\; - {u_l}\frac{{\partial g_{l,m}^n}}{{\partial x}} - {v_m}\frac{{\partial g_{l,m}^n}}{{\partial y}} + \frac{{{g^ + } - g}}{{{\tau ^n}}} = \\ \;\;\;\;\; - R + \frac{{{g^ + } - g}}{{{\tau ^n}}} \end{array}$ (3)

 $R = \frac{{\int_0^{\Delta T} {\sum\limits_{ii = 1}^4 {{u_{\rm{n}}}{g_{\rm{p}}}} } (t)dt}}{{\Delta T}}$ (4)

 $\Delta T < \Delta {t_{\min }}/CFL = \Delta {T_{{\rm{ref}}}}$ (5)

 $\begin{array}{l} (1 + \Delta t/{\tau ^n}){(\Delta g)_{i,j,l,m}} + \frac{{\Delta t}}{{\left| {{V_{i,j}}} \right|}}\sum\limits_{ii = 1}^4 {({u_{l,m}} \cdot {n_{ii}})} \cdot \left| {{S_{i,j,ii}}} \right| \cdot \\ \;\;\;\;\;\;FF({(\Delta g)_{i,j,l,m}},{(\Delta g)_{i1,j1,l,m}}) = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\Delta t \cdot L_{i,j,l,m}^n \end{array}$ (6)

 $\begin{array}{*{20}{l}} {FF({{(\Delta g)}_{i,j,l,m}},{{(\Delta g)}_{i1,j1,l,m}}) = }\\ {\;\;\;\frac{1}{2}({{(\Delta g)}_{i,j,l,m}} + {{(\Delta g)}_{i1,j1,l,m}}) + }\\ {\;\;\;\;\frac{1}{2}{\rm{sign}}({u_{l,m}} \cdot {\rm{ }}{n_{ii}})({{(\Delta g)}_{i,j,l,m}} - {{(\Delta g)}_{i1,j1,l,m}})} \end{array}$

 $\begin{array}{l} {(\Delta g)_{i,j,l,m}} + \sum\limits_{ii = 1,2,3,4} {\Delta t \cdot {z_{i,j,l,m}} \cdot {{(\Delta g)}_{i1,j1,l,m}}} = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\frac{{\Delta t}}{{{\lambda _{i,j,l,m}}}} \cdot L_{i,j,l,m}^n \end{array}$ (7)

 $\begin{array}{*{20}{l}} {{\lambda _{i,j,l,m}} = 1 + \Delta t/{\tau ^n} + \Delta t \cdot {b_{i,j,l,m}},{z_{i,j,j,m}} = \frac{{{c_{i,j,l,m}}}}{{{\lambda _{i,j,l,m}}}}}\\ {{b_{i,j,l,m}} = {\rm{ }}\sum\limits_{ii = 1}^4 ( {u_{l,m}} \cdot {\rm{ }}{n_{ii}}) \cdot (1 + {\rm{sign}}({u_{l,m}} \cdot {\rm{ }}{n_{ii}}))\frac{{\left| {{S_{i,j,ii}}} \right|}}{{2\left| {{V_{i,j}}} \right|}}}\\ {{c_{i,j,l,m}} = ({u_{l,m}} \cdot {\rm{ }}{n_{ii}}) \cdot (1 - {\rm{sign}}({u_{l,m}} \cdot {n_{ii}}))\frac{{\left| {{S_{i,j,ii}}} \right|}}{{2\left| {{V_{i,j}}} \right|}}} \end{array}$

 $\begin{array}{*{20}{l}} {(I + \Delta t \cdot {Z_{l,m}}) \cdot {{(\Delta g)}_{l,m}} = \Delta t \cdot \Lambda _{l,m}^{ - 1} \cdot {\rm{ }}L_{l,m}^n}\\ {{{(\Delta g)}_{l,m}} = \left( {\begin{array}{*{20}{c}} {{{(\Delta g)}_{1,1,l,m}}}\\ {{{(\Delta g)}_{2,1,l,m}}}\\ \cdots \\ {{{(\Delta g)}_{NI - 1,NJ - 1,l,m}}} \end{array}} \right),{\rm{ }}L_{l,m}^n = \left( {\begin{array}{*{20}{c}} {L_{1,1,l,m}^n}\\ {L_{2,1,l,m}^n}\\ \cdots \\ {L_{NI - 1,NJ - 1,l,m}^n} \end{array}} \right)}\\ {{\Lambda _{l,m}} = \left( {\begin{array}{*{20}{c}} {{\lambda _{1,1,l,m}}}&0& \cdots &0\\ 0&{{\lambda _{2,1,l,m}}}& \cdots &0\\ 0&0& \cdots &0\\ 0&0& \cdots &{{\lambda _{NI - 1,NJ - 1,l,m}}} \end{array}} \right)} \end{array}$ (8)

 $I + \Delta t \cdot Z_{l,m} = L_{l,m} \cdot U_{l,m} + O(\Delta t^2)$

$L_{l,m}$和$U_{l,m}$都是标量三对角矩阵

 $\begin{array}{l} {l_{pq}} = \left\{ {\begin{array}{*{20}{l}} {\Delta t \cdot {z_{pq}},}&{p < q}\\ {0,}&{p > q} \end{array}} \right.\\ {u_{pq}} = \left\{ {\begin{array}{*{20}{l}} {0,}&{p < q}\\ {\Delta t \cdot {z_{pq}},}&{p > q} \end{array}} \right.\\ {l_{pp}} = {u_{pp}} = 1 \end{array}$

 ${L_{l,m}} \cdot {U_{l,m}} \cdot {(\Delta g)_{l,m}} = \Delta t \cdot {\rm{ }}\Lambda _{l,m}^{ - 1} \cdot L_{l,m}^n$ (9)

 $f^ + = f^ + ({\rho }',{U}',{V}',{\lambda }',{q}'_x ,{q}'_y )$

 $\begin{array}{l} \sum {{f^ + }\psi _2^{\rm{T}}} - {\left( {\rho ,\rho U,\rho V,\rho E,{q_x},{q_y}} \right)^{\rm{T}}} = \\ \;\;\;\;\;\;\;\;\;\;\;{\left( {0,0,0,0,- {\mkern 1mu} 2/3{q_x},- {\mkern 1mu} 2/3{q_y}} \right)^{\rm{T}}} \end{array}$ (10)

2 数值结果

 图 1 计算收敛以后驻点线上第二层网格界面上密度通量随演化时间的变化 Fig.1 The time-dependent flux of density on the secondinterface along the stagnation line
 图 2 计算收敛以后驻点线上第二层网格界面上时间平均的密度通量随演化时间的变化 Fig.2 The time-averaged flux of density on the secondinterface along the stagnation line

 图 3 不同网格下圆柱表面应力分布 Fig.3 Stress on the solid surface
 图 4 不同网格下圆柱表面应力分布 Fig.4 Stress on the solid surface

 图 5 阻力和残差收敛历程 Fig.5 Convergent histories of the drag coefficient and residual
 图 6 阻力和残差收敛历程 Fig.6 Convergent histories of the drag coefficient and residual

 图 7 圆柱算例, 压力分布 Fig.7 Pressure contour around the cylinder
 图 8 圆柱表面压力比较 Fig.8 Pressure distribution on the cylinder wall
 图 9 圆柱表面热流比较 Fig.9 Heat flux on the cylinder wall
 图 10 圆柱表面应力比较 Fig.10 Stress distribution on the cylinder wall

 图 11 尖楔算例, 压力分布 Fig.11 Pressure contour around the wedge
 图 12 尖楔表面压力比较 Fig.12 Pressure distribution on the wedge wall
 图 13 尖楔表面热流比较 Fig.13 Heat flux on the wedge wall
 图 14 尖楔表面应力比较 Fig.14 Stress distribution on the wedge wall
3 结 论

(1)对高速流动定常问题,隐式UGKS具有明显的加速收敛效果,与显式方法相比,计算效率提高1$\sim$2个量级;

(2)UGKS与DS2V在高超声速圆柱和尖楔上的计算结果高度吻合,进一步验证了隐式方法实现的正确性以及UGKS方法在模拟高超声速流动中良好的精度.

 [1] Yu PB. A unied gas kinetic scheme for all knudsen number flows. [PhD Thesis], Hong Kong:The Hong Kong University of Science and Technology,2013. [2] Chen SZ, Xu K, Lee CB, et al. A unified gas kinetic scheme with moving mesh and velocity space adaptation. Journal of Computational Physics, 2012, 231(20): 6643-6664. [3] Arslanbekov RR, Kolobov VI, Kolobov AA. Kinetic solvers with adaptive mesh in phase space, In: Abe T, Ed. Rarefied Gas Dynamics, Melville, 2012, Amer Inst Physics, 2012, 294-301. [4] Baranger C, Claudel J, Claudel N, et al. Locally refined discrete velocity grids for deterministic rarefied flow simulations, In: Abe T, Ed. Rarefied Gas Dynamics, Melville, 2012, Amer Inst Physics, 2012, 389-396. [5] Yoon S, Jameson A. Lower upper symmetric Gauss Seidel method for the Euler and Navier-Stokes equations. AIAA Journal, 1988, 26(efeq9): 1025-1026. [6] Yang JY, Huang JC. Rarefied flow computations using nonlinear model Boltzmann equations. Journal of Computational Physics, 1995, 120(efeq2): 323-339. [7] 毕林. Boltzmann模型方程数值算法在槽道流问题中的应用研究. [硕士论文].四川绵阳: 中国空气动力研究与发展中心, 2007 (Bi Lin. Application and study on numerical algorithm of the Boltzmann model equation in channel flow. [Master Thesis]. Mianyang: China Aerodynamic Research and Development Center, 2007 (in Chinese)) [8] 王强, 程晓丽, 庄逢甘. 基于简化Boltzmann方程的超声速流稀薄效应分析. 航空学报, 2005. 26(efeq3):281-285 (Wang Qiang, Cheng Xiaoli, Zhuang Fenggan. Rarefaction effect analysis of supersonic flows based on reduced Boltzmann equations. Acta Aeronautica Et Astronautica Sinica, 2005, 26(efeq3): 281-285 (in Chinese)) [9] 王强, 程晓丽, 庄逢甘. 稀薄流非线性模型方程离散速度坐标法有限差分解. 计算力学学报, 2006, 23(efeq2): 235-241 (Wang Qiang, Cheng Xiaoli, Zhuang Fenggan. Finite difference solution of nonlinear model equations for rarified gas using discrete velocity ordinate method. Chinese Journal of Computational Mechanics, 2006, 23(efeq2): 235-241 (in Chinese)) [10] Titarev VA. Efficient deterministic modelling of three-dimensional rarefied gas flows. Communications in Computational Physics, 2012, 12(efeq1):162-192. [11] Titarev VA, Dumbser M, Utyuzhnikov S. Construction and comparison of parallel implicit kinetic solvers in three spatial dimensions. Journal of Computational Physics, 2014, 256: 17-33. [12] Xu K, Huang JC. A unified gas-kinetic scheme for continuum and rarefied flows. Journal of Computational Physics, 2010, 229(20): 7747-7764. [13] Huang JC, Xu K, Yu PB. A unified gas-kinetic scheme for continuum and rarefied flows II: multi-dimensional cases. Communications in Computational Physics, 2012, 12(efeq3): 662-690. [14] Huang JC, Xu K, Yu PB. A unified gas-kinetic scheme for continuum and rarefied flows III: Microflow simulations. Communications in Computational Physics, 2012, 14(efeq5): 1147-1173. [15] 徐昆, 李启兵, 黎作武. 离散空间直接建模的计算流体力学方法. 中国科学: 物理学 力学 天文学, 2014, 44(efeq5):519-530 (Xu Kun, Li Qibing, Li Zuowu. Direct modeling-based computational fluid dynamics. Scientia Sinica Physica, Mechanica & Astronomica, 2014, 44(efeq5): 519-530 (in Chinese)) [16] Xu K. A gas-kinetic BGK scheme for the Navier-Stokes equations and its connection with artificial dissipation and Godunov method. Journal of Computational Physics, 2001, 171(efeq1): 289-335. [17] 毛枚良, 徐昆, 邓小刚. 动能BGK算法在近连续流模拟中的应用. 空气动力学学报, 2005, 23(efeq3): 317-321 (Mao Meiliang, Xu Kun, Deng Xiaogang. Application of kinetic BGK algorithm in simulating near continuum flow. Acta Aerodynamica Sinica, 2005, 23(efeq3): 317-321 (in Chinese)) [18] Xu K, Mao ML, Tang L. A multidimensional gas-kinetic BGK scheme for hypersonic viscous flow. Journal of Computational Physics, 2005, 203(efeq2): 405-421. [19] Jiang J, Qian YH. Implicit gas-kinetic BGK scheme with multigrid for 3D stationary transonic high-Reynolds number flows. Computers & Fluids, 2012, 66: 21-28. [20] Li WD, Masayuki K, Kazuhiko S. An implicit gas kinetic BGK scheme for high temperature equilibrium gas flows on unstructured meshes. Computers & Fluids, 2014, 93: 100-106. [21] Jiang DW, Li J, Mao ML, et al. Implicit implementation of bgk-ns method and its application in complex flows. Transactions of Nanjing University of Aeronautics & Astronautics, 2013, 30(supp): 87-92. [22] Bird GA. The DS2V/3V program suite for DSMC calculations, In: Capitelli M, Ed. Rarefied Gas Dynamics, 2005, Amer Inst Physics, 2005, 541-546. [23] Shakhov E. Generalization of the Krook kinetic equation. Fluid Dynamics, 1968, 3(efeq5): 95-96. [24] 江定武, 毛枚良, 李锦 等. 气体动理学统一算法中相容性条件不满足引起的数值误差及其影响研究. 力学学报, 2014, 47(efeq1): 163-168 (Jiang Dingwu, Mao Meiliang, Li Jin, et al. Study on the numerical error introduced by dissatisfying the conservation constraint in UGKS and its effects. Chinese Journal of Theoretical and Applied Mechanics, 2014, 47(efeq1): 163-168 (in Chinese)) [25] Liu HL, Cai CP, Zou C. An object-oriented serial implementation of a DSMC simulation package. Computers & Fluids, 2012, 57: 66-75.
STUDY ON IMPLICIT IMPLEMENTATION OF THE UNIFIED GAS KINETIC SCHEME
Mao Meiliang, Jiang Dingwu, Li Jin, Deng Xiaogang
1. State Key Laboratory of Aerodynamics , China Aerodynamics Research and Development Center, Mianyang 621000, China;
2. Computational Aerodynamics Institute, China Aerodynamics Research and Development Center, Mianyang 621000, China;
3. National University of Defense Technology, Changsha 410073, China
Fund: The project was supported by the National Natural Science Fundation of China(11402287) and Study Fund of State Key Laboratory of Aerodynamics (SKLA2015-1-2).
Abstract: The current explicit unified gas kinetic scheme (UGKS) is very time-consuming for high speed flows due to the massive phase space mesh demand, which is a bottleneck for complex engineering problems. In order to improve the efficiency, the motion and collision term in the model equation is implicitly treated and the evolving time averaged flux across the cell interface is introduced, then the implicit UGKS can be obtained applying the approximate LU decomposition on the matrix of the governing equations. The tests on the flows over a cylinder with different freestream Mach numbers show that the implicit method can give the same result as the original explicit method with a properly chosen evolving time step. Meanwhile, the computational efficiency can be improved by 1-2 orders.
Key words: unified gas kinetic scheme    implicit method    convergence acceleration